In any subsurface hydrocarbon exploration and development, implied measurements such as detailed geological description and outcrop data, and specific measurements such as pressure transient, seismic, cores, logs, and fluid samples provide useful information for static and dynamic reservoir characterization, development of simulation models, and forecasting. However, core and log data delineate rock properties only in the vicinity of the wellbore while geological and seismic data usually are not directly related to formation transport properties such as permeability. Pressure transient data from drill stem testing (DST) or permanent downhole pressure sensors, production, and wireline formation tests provide dynamic data such as reservoir pressure and flow rate that can be used to estimate formation rock properties and fluid distributions and for dynamic reservoir description. Therefore, such tests are very useful for exploration environments and field development and reservoir management as well as for general production and reservoir engineering.
Conventional well tests such as DST have traditionally been used to obtain spatial distributions of the formation permeability and reservoir pressure based on the history matching of the pressure data to an analytical or a numerical model selected to best represent the flow regimes observed from diagnostic plots. In this application, this is referred to as a “lumped average” approach.
The most well-known lumped average techniques include the simple analytical model where the lumped parameters are mainly permeability-thickness product, permeability, skin factor, wellbore storage co-efficient and fracture length, etc. Recently, non-linear least-squares optimization has been applied to pressure transient data using numerical models with a similarly limited number of parameters.
As the need for more spatial resolution of the parameters increases, researchers have turned to “pixel” methods where the physical properties of the reservoir are discretized on a regular pixel-like grid over the reservoir domain. Such pixel based approaches have received considerable attention in the petroleum literature. The following publications disclose the application of techniques where dense geological information is used as a prior and regularizing scheme for an inversion:    Abacioglu, Y., Reynolds, A. C., and Oliver, D. S. (1997), “Estimating Heterogeneous Anisotropic Permeability Fields from Multiwell Interference Tests: A Field Example,” 1997 SPE Annual Technical Conference and Exhibition, number SPE 38654, San Antonio, Tex., U.S.A.    Chu, L., Reynolds, A. C., and Oliver, D. S. (1995), “Reservoir Description From Static and Well-Test Data Using Efficient Gradient Methods,” International Meeting on Petroleum Engineering, number SPE 29999, Beijing, P.R. China.    He, N., Reynolds, A., and Oliver, D. S. (1996), “Three-Dimensional Reservoir Description from Multiwell Pressure Data and Prior Information,” 1996 SPE Annual Technical Conference and Exhibition, number SPE 36509, Denver, Colo., U.S.A.    He, N., Oliver, D. S., and Reynolds, A. C. (1997), “Conditioning Stochastic Reservoir Models to Well-Test Data,” 1997 SPE Annual Technical Conference and Exhibition, number SPE 38655, San Antonio, Tex., U.S.A.    Oliver, D. S. (1996), “Multiple Realizations of the Permeability Field from Well Test Data,” SPE Journal, June: 145-154.    Oliver, D. S., Reynolds, A. C., & Liu, N. (2008). Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge: Cambridge University Press.    Reynolds, A., He, N., Chu, L., and Oliver, D. (1996), “Reparameterization Techniques for Generating Reservoir Descriptions Conditioned to Variograms and Well-test Pressure Data,” SPE Annual Technical Conference and Exhibition, number SPE 30588, Dallas, Tex., U.S.A.
The following publications have considered the inversion of production data on pixel-like grids:    Bi, Z., Oliver, D., and Reynolds, A. (2000). “Conditioning 3D Stochastic Channels To Pressure Data,” Society of Petroleum Engineers Journal, December (4): 474-484.    Landa, J. L., Kamal, M. M., Jenkins, C. D., and Horne, R. N. (1996). “Reservoir Characterization Constrained to Well Test Data: A Field Example,” 1996 SPE Annual Technical Conference and Exhibition, number SPE 36511, Denver, Colo., U.S.A.    Phan, V. Q. and Horne, R. N. (2002). Fluvial channel parameter estimation constrained to static, production, and 4D seismic data. In SPE Annual Technical Conference and Exhibition, number SPE 77518, San Antonio, Tex., U.S.A.
An alternative method of optimization for geostatistical parameters such as the correlation length and variance is discussed in:    Gautier, Y. and Noetinger, B. (1998). “Determination of Geostatistical Parameters Using Well Test Data,” In SPE Annual Technical Conference and Exhibition, number SPE 49278, New Orleans, La., U.S.A.    Yadavalle, S. K., Roadifer, R. D., Jones, J. R., and Yeh, N.-S. (1994). “Use of Pressure Transient Data to Obtain Geostatistical Parameters For Reservoir Characterisation,” 69th Annual Technical Conference and Exhibition, number SPE 28432, pages 719-732, New Orleans, La., U.S.A.
The following publications disclose ensemble Kalman filtering techniques applied to pixel-like grids:    Aanonsen, S. I., Naevdal, G., Oliver, D. S., Reynolds, A. C. and Valles, B. (2009). Review of ensemble Kalman filter in petroleum engineering (SPE 117724), SPE Journal, 14(3), 393-412.    Evensen, G. (2007). Data Assimilation: The Ensemble Kalman Filter, Springer, Berlin.
The following publications should be considered as background art related to sensitivity and uncertainty analysis in pixel-based methods:    Booth, R. J. S., Morton, K. L., Onur, M., Kuchuk, F. J. (2010). Grid-based Inversion of Pressure Transient Test Data, presented at 12th European Conference on the Mathematics of Oil Recovery, Oxford, UK, 6-9 September (incorporated herein by reference).    Farmer, C. L. (2007). Bayesian field theory applied to scattered data interpolation and inverse problems; Algorithms for Approximation, 147-166.